MathDB

We number a hundred blank cards on both sides with the numbers

1

100

. The cards are then stacked in order, with the card with the number

1

on top. The order of the cards is changed step by step as follows: at the

1

st step the top card is turned around, and is put back on top of the stack (nothing changes, of course), at the

2

nd step the topmost

2

cards are turned around, and put back on top of the stack, up to the

100

th step, in which the entire stack of

100

cards is turned around. At the

101

st step, again only the top card is turned around, at the

102

nd step, the top most

2

cards are turned around, and so on. Show that after a finite number of steps, the cards return to their original positions.

Problem Statement